As an alternative to @SteveV's strategy, there is another way to approach this puzzle, by considering:
numbers whose names can be reduced by removing letters to leave the letter 'E' or the digraph 'EE', which are the hexadecimal equivalents of the decimal values '14' and '238', respectively, i.e. even numbers! (There's nothing in the original question to preclude using base systems other than decimal, after all...)
Using this method, it is possible to use even smaller odd numbers than in that answer, right across the board:
Removing 2 letters: ONE → E (14)
3: THREE → EE (238)
4: THREE → E (14)
5: ELEVEN → E (14)
6: THIRTEEN → EE (238)
7: THIRTEEN → E (14)
8: SEVENTEEN → E (14)
9: TWENTY THREE → EE (238)
(NB There is no point considering the letters A, B, C or D, which are valued 10-13 in hexadecimal, since there are no English odd numbers less than 100 - a stipulation of the original puzzle - that contain these letters... You'll need to count as high as one-hundred-and-one, one-billion-and-one, or one-octillion-and-one before you find them! And while we could consider combinations involving the letter 'F' (15) also, the fact that it is rarer than the letter 'E' in the names of small odd numbers actually means there is always a smaller solution using 'E's alone...)
Note that it is impossible to use this same technique to find a solution requiring the removal of just one letter...
Unless we 'up our game' even further and switch from using hexadecimal to using a higher base, like base-24! Then we can legally do:
Removing 1 letter: ONE → NE (566 in base-24)